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Properties

Label 2-252-252.139-c1-0-5
Degree $2$
Conductor $252$
Sign $0.923 - 0.383i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 − 0.0664i)2-s + (−0.758 − 1.55i)3-s + (1.99 + 0.187i)4-s + (2.66 + 1.53i)5-s + (0.968 + 2.24i)6-s + (1.15 + 2.37i)7-s + (−2.80 − 0.397i)8-s + (−1.84 + 2.36i)9-s + (−3.66 − 2.35i)10-s + (−3.64 + 2.10i)11-s + (−1.21 − 3.24i)12-s + (2.97 + 1.71i)13-s + (−1.47 − 3.43i)14-s + (0.374 − 5.31i)15-s + (3.92 + 0.747i)16-s + 2.29i·17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0469i)2-s + (−0.437 − 0.898i)3-s + (0.995 + 0.0938i)4-s + (1.19 + 0.688i)5-s + (0.395 + 0.918i)6-s + (0.437 + 0.899i)7-s + (−0.990 − 0.140i)8-s + (−0.616 + 0.787i)9-s + (−1.15 − 0.743i)10-s + (−1.10 + 0.635i)11-s + (−0.351 − 0.936i)12-s + (0.824 + 0.476i)13-s + (−0.394 − 0.918i)14-s + (0.0966 − 1.37i)15-s + (0.982 + 0.186i)16-s + 0.555i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.923 - 0.383i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.923 - 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833649 + 0.166019i\)
\(L(\frac12)\) \(\approx\) \(0.833649 + 0.166019i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0664i)T \)
3 \( 1 + (0.758 + 1.55i)T \)
7 \( 1 + (-1.15 - 2.37i)T \)
good5 \( 1 + (-2.66 - 1.53i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.64 - 2.10i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.97 - 1.71i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.29iT - 17T^{2} \)
19 \( 1 - 1.70T + 19T^{2} \)
23 \( 1 + (-4.68 - 2.70i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.76 + 8.25i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.22 + 5.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.04T + 37T^{2} \)
41 \( 1 + (4.43 + 2.56i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.89 + 1.67i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.20 - 3.82i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + (-4.44 + 7.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.57 - 2.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.53 + 5.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.48iT - 71T^{2} \)
73 \( 1 + 3.46iT - 73T^{2} \)
79 \( 1 + (8.87 - 5.12i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.0572 + 0.0991i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.61iT - 89T^{2} \)
97 \( 1 + (5.53 - 3.19i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.83657060948203027037945360690, −11.09043128011520478773059530782, −10.23373301851045916794977891103, −9.234409830674547830847019374602, −8.143662578256421112499429035678, −7.22195805270602316619836586128, −6.12981492034690929320540297414, −5.53945202384906839056899151714, −2.58178812642825375025938471253, −1.80591319103876534329988202565, 1.05391511912231731816956626562, 3.12154160524971795202615149611, 5.06451001098294166727631386683, 5.73769507037984823937057780297, 7.07252425071952009467989323905, 8.497914971969380895567330606856, 9.086118311962153409750308914936, 10.34967168328204481474449679014, 10.51404339277783698531336281275, 11.52517406419599144308102136551

Graph of the $Z$-function along the critical line